IMC 2009 · Problem 1

Suppose that $f$ and $g$ are real-valued functions on the real line and $f(r) \le g(r)$ for every rational $r$. Does this imply that $f(x) \le g(x)$ for every real $x$ if

a) $f$ and $g$ are non-decreasing?

b) $f$ and $g$ are continuous?